Classification of Nonlinear Vibrations in Symmetric Molecules : Equivariant Degree Method




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This dissertation is devoted to the study of nonlinear periodic vibrations of a group of particles within symmetric molecular conformations governed by the Lennard-Jones and Coulomb forces. In particular, the author investigates nonlinear vibrational modes of oscillations for molecules with tetrahedral and dihedral configurations of atoms. Using the gradient equivariant degree, the author provides the full topological classification of the periodic solutions with both temporal and spatial symmetries for the above configurations. In the process, the general formulae for processing the equivariant spectral data of the linearized system and obtaining the critical frequencies of the particle motions are devised. The obtained frequencies constitute the set of all critical periods for small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions. The evaluated equivariant invariants provide a complete list of the spatio-temporal symmetries of the periodic solutions emerging from the ground state symmetric equilibria. The proposed method can be applied to study nonlinear vibrations in other symmetric molecules with general force fields.



Vibration, Nonlinear oscillations, Atoms


©2019 Irina Berezovik. All rights reserved.